Due to the uncertainty of the obtained eigenvalues and eigenfunctions, the estimation and its confidence intervals can be corrected through bootstrap in MFPCA. We generalize the corrected fitting and confidence intervals from the univariate ([1]} [1]}) to the multivariate case from \(B\)  times bootstrap. Specifically, the fit is obtained through the iterated expectation, and the confidence interval is obtained through the iterated variance. We name the bootstrap corrected estimation the BMFPCA fit:
\(\widehat{\mathbf {X}}_i={\textrm {E}}_{\widehat{\mathbf {\theta }}}\Big \lbrace {\textrm {E}}_{{\mathbf {Y}}_i|\widehat{\mathbf {\theta }}}\Big [{\mathbf {Y}}_i|\widehat{\mathbf {\theta }}\Big ]\Big \rbrace ,~~~i=1,\ldots ,n.\) 
